For clarity of exposition we shall carry out our construction for the 2-adic odometer; i.e. \(q_n=2\)  for all \(n\in \mathbb {N}\) . The construction for other odometers is analogous. Let \(\mathbb {D}^2:=\lbrace z\in \mathbb {C}, |z|\le 2 \rbrace \) . We let \(\varphi :\mathbb {D}^2\rightarrow \mathbb {D}^2\)  be the branched 2-fold covering given by
\(\varphi (r\cos (\theta ), r\sin (\theta ))=(r\cos (2\theta ), r\sin (2\theta )).\) 
Let \(\bar{0}\)  be the complex zero, \(\@root 2^n \of {1}=\lbrace w_j:j=0,\ldots ,2^n-1\rbrace \)  be the complex \(2^n\) -th roots of 1, and \(\ell _l:=\lbrace (r\cos (\frac{l2\pi }{2^n}), r\sin (\frac{l2\pi }{2^n}))|r\in [0,1]\rbrace \)  be the chords from \(\bar{0}\)  to \(w_l\) , for \(l=0,\ldots ,2^{n}-1\) .
We set
\(T_n:=\bigcup _{i=0}^{2^{n}-1}\ell _i\) 
and note that \(T_n\)  is invariant under rotation
\(R_n:\mathbb {D}^2\rightarrow \mathbb {D}^2\)  defined by
\(R_n(r\cos (\theta ), r\sin (\theta ))=(r\cos (\theta +2^{-n+1}\pi ), r\sin (\theta +2^{-n+1}\pi )).\) 
The following claim is implicitly contained in the proof of [1]}.
Claim 4.1 
For any integer \(n> 0\)  and any \(\gamma >0\) , there exist map \(q_{\gamma ,n}\colon T_n\rightarrow [0,1]\) ,
and family of maps \((\hat{f}_{\gamma ,n}:T_n\rightarrow T_n: n\in \mathbb {N})\)  so that \(\hat{f}_{\gamma ,n}\)  is piecewise linear and topologically exact, invariant under rigid rotation of branches of \(T_n\) , and such that \(\hat{f}_{\gamma ,n-1}\circ \varphi =\varphi \circ \hat{f}_{\gamma ,n}\)  and if \(q_{\gamma ,n}(x)=q_{\gamma ,n}(y)\)  then \(d(\hat{f}_{\gamma ,n}(x), \hat{f}_{\gamma ,n}(y))<\gamma \) .

Now we will prove the core claim of this proof, which uses also the claim above.
Claim 4.2 
There exists a family of maps \(g_{i,n}:T_n\rightarrow T_n\)  such that if we set \(f_{i,n}=\hat{f}_{i,n}\circ g_{i,n}\)  then \(\mathcal {P}_{n}=\varprojlim (T_{n}, f_{i,n})\)  is a pseudo-arc for each \(n\) .

The construction will be “diagonal”.
Let \(\lbrace \varepsilon _k\rbrace _{k=0}^\infty \)  be any sequence of real numbers decreasing to 0. Later, while constructing maps \(f_{k,n}\) , we will speed up convergence of the sequence \(\varepsilon _k\)  to satisfy assumptions of Lemma REF .
Set \(\gamma _0=\varepsilon _0\) .
Let \(\hat{f}_{0,1}=\hat{f}_{\gamma _0,1}\)  and \(q_{0,1}=q_{\gamma _0,1}\)  be provided by Lemma REF 
for \(\gamma =\gamma _0\) . Since \(\hat{f}_{0,1}\)  is topologically exact
we can find an integer \(s_0>0\)  such that \(\hat{f}_{0,1}^{s_0}(\ell _0)=\hat{f}_{0,1}^{s_0}(\ell _1)=T_1\) .
Next, applying Remark REF  to \(\hat{f}_{0,1}^{s_0}\)  we can find a map \(h_{0,1}\)  and a positive integer \(j_0\)  such that if we denote \(F_{0,1}=\hat{f}_{0,1}^{s_0}\circ h_{0,1}\) , then each map \(\omega \colon [0, 1] \rightarrow T_n\)  is \((F^{j_0}_{0,1}, \varepsilon _0)\) -crooked.
Denote \(f_{0,1}=F^{j_0}_{0,1}\)  and observe that we have \(f_{0,1}=\hat{f}_{0,1}\circ g_{0,1}\) , where \(g_{0,1}=\hat{f}_{0,1}^{s_0-1}\circ h_{0,1}\circ F^{j_0-1}_{0,1}\) .
Next, assume that the maps \(f_{i,n},q_{i,n},h_{i,n},g_{i,n}\)  have already been defined for \(n=1,\ldots ,k\)  and \(0\le i \le k-1\) . We adjust \(\varepsilon _{k+1}\) , if necessary, to be so small that
\(\varepsilon _{k+1}<\min _{n\le k;i<k}L(2^{-k-1}\operatorname{diam}(T_n), f_{i,n}\circ f_{i+1,n}\circ \ldots \circ f_{k-1,n}).\) 
For each \(0\le j \le k\)  we let \(f_{j,k+1}\)  be a lift of \(f_{j,k}\)  through the branched cover \(\varphi \)  and similarly we define maps \(g_{j,k+1}, h_{j,k+1}\) .
Let \(\gamma _{k+1}<\varepsilon _{k+1}\)  be such that, for each \(n=1,\ldots , k+1\) , if \(x,y\in T_n\)  are such that \(d(x,y)<\gamma _{k+1}\) ,
then for each \(0\le j< k\) 
we have
\(d(f_{j,n}\circ \ldots \circ f_{k-1,n}(x),f_{j,n}\circ \ldots \circ f_{k-1,n}(y))<\varepsilon _{k+1}\) 
and
\(d(g_{j,n}\circ f_{j+1,n}\circ \ldots \circ f_{k-1,n}(x),g_{j,n}\circ f_{j+1,n}\circ \ldots \circ f_{k-1,n}(y))<\varepsilon _{k+1}.\) 
Apply Lemma REF  to \(\gamma _{k+1}\)  to get maps
\(\hat{f}_{k,n}=\hat{f}_{\gamma _{k+1},n}\)  and \(q_{k,n}=q_{\gamma _{k+1},n}\)  for \(n=1,\ldots , k+1\) . By topological exactness, we select a positive integer \(s_{k}\) 
such that \(\hat{f}_{k,n}^{s_k}(\ell _i)=T_n\)  for \(i=0,\ldots ,n-1\) .
Repeating the argument from above, by Remark REF  we can find a map \(h_{k,k+1}\)  and a positive integer \(j_{k+1}\)  such that for the map \(F_{k,k+1}=\hat{f}_{k,k+1}^{s_k}\circ h_{k,k+1}\)  each map \(\omega \colon [0, 1] \rightarrow T_{1}\)  is \((F^{j_{k+1}}_{k,k+1}, \varepsilon _{k+1})\) -crooked.
For each \(j=0,1,\ldots ,k\)  we let \(h_{k,k-j}\)  to be a projection of \(h_{k,k+1-j}\)  through the branched cover \(\varphi \) , which is well defined since \(h_{k,k+1}\)  is invariant under rotation of branches of \(T_{k+1}\)  (see Remark REF ).
Denote \(F_{k,n}=\hat{f}_{k,n}^{s_k}\circ h_{k,n}\)  for \(n\le k+1\)  and
observe that we can represent \(f_{k,n}=\hat{f}_{k,n}\circ g_{k,n}\) , where \(g_{k,n}= \hat{f}_{k,n}^{s_k-1}\circ h_{k,n}\circ F^{j_{k+1}-1}_{k,n}\) .
Since \(\varphi \)  does not increase distance, and for every \(n\le k\)  each \(\omega \colon [0,1]\rightarrow T_n\)  can be represented as \(\omega =\varphi \circ \omega ^{\prime }\)  for some \(\omega ^{\prime }\colon [0,1]\rightarrow T_{n+1}\) , we see that each such map \(\omega \)  is \((F^{j_{k+1}}_{k,n}, \varepsilon _{k+1})\) -crooked.
By the construction, Lemma REF  ensures that each \(\mathcal {P}_n:=\varprojlim (T_n,f_{i,n})\)  is hereditarily indecomposable. We also have an \(2^n\) -periodic homeomorphism \(H\colon \mathcal {P}_n \rightarrow \mathcal {P}_n\)  induced by \(2^n\) -periodic rotations of \(T_n\)  on each coordinate, where \(\mathcal {P}_n=\varprojlim (T_n,f_{i,n})\) .
It remains to prove that each \(\mathcal {P}_{n}\)  is arc-like.
In order to prove it we note that \(\mathcal {P}_n=\varprojlim (T_n,r_{j,n})\) 
where \(r_{2j,n}=\hat{f}_{j,n}\)  and \(r_{2j+1,n}=g_{j,n}\)  for \(j=1,2,\ldots \) .
But then, if \(\pi _{j,n}\)  denotes the \(j\) -th coordinate projection from \(\mathcal {P}_n\)  to \(T_n\) , we have a well defined projection \(p_{j,n}=q_{j,n}\circ \pi _{2j,n}\colon \mathcal {P}_n\rightarrow [0,1]\) 
with the property that if \(p_{k,n}(\hat{x})=p_{k,n}(\hat{y})\)  for some \(k\)  and \(\hat{x},\hat{y}\in \mathcal {P}_n\) , then
\(d(r_{2k,n}(\pi _{2k,n}(\hat{x})),r_{2k,n}(\pi _{2k,n}(\hat{y})))=d(\hat{f}_{k,n}(\pi _{2k,n}(\hat{x})),\hat{f}_{k,n}(\pi _{2k,n}(\hat{y})))<\gamma _k\) 
which by (REF ) and (REF ) implies
that for \(j=0,\ldots , 2k-1\)  we have
\(d(\pi _{j,n}(\hat{x}),\pi _{j,n}(\hat{y}))<\varepsilon _k.\) 
But then
\(d(\hat{x},\hat{y})\le \varepsilon _k\sum _{j=0}^{2k-1}2^{-j}+\sum _{j=2k}^{\infty }2^{-j}\le 2\varepsilon _k+2^{-2k+1}.\) 
Therefore, for every \(\varepsilon >0\)  there exists a natural number \(k\)  so that \(p_{k,n}\)  is an \(\varepsilon \) -map.
This shows that for all \(n\ge 1\)  the space \(\mathcal {P}_n\)  is arc-like and thus \(\mathcal {P}_n\)  is the pseudo-arc, which proves the claim.
For what follows we refer the reader to the Diagram .
Since \(T_1\)  is an arc, by [2]} the maps \((f_{i,1}:i\in \mathbb {N})\)  extend to near-homeomorphisms \((\bar{f}_{i,1}:i\in \mathbb {N})\)  of the topological disk \(\mathbb {D}^2\)  (see also [3]}). Then, since \(f_{i,n}\circ \varphi =\varphi \circ f_{i,n+1}\)  for all \(i\in \mathbb {N}\)  and \(n\ge 1\) , all the maps \((f_{i,n}:i\in \mathbb {N},n\ge 1)\)  extend to near-homeomorphisms \((\bar{f}_{i,n}:i\in \mathbb {N},n\ge 1)\)  of \(\mathbb {D}^2\) .
Consequently each \(\mathcal {P}_{n}\)  is embedded in \(\mathbb {D}^2\)  by Brown's theorem [4]}.
Now let the homeomorphisms \(\hat{R}_n:S_n\rightarrow S_n\) , where \(S_n=\varprojlim (\mathbb {D}^2,f_{i,n})\supset \mathcal {P}_n\)  for each \(n\ge 1\) , and \(\hat{R}_\infty :\mathbb {D}_\infty \rightarrow \mathbb {D}_\infty \) , where \(\mathbb {D}_{\infty }=(S_1,S_2,S_3,\ldots )\) , be given as follows
\(\hat{R}_n=(R_n,R_n,R_n,\ldots ),\) 
\(\hat{R}_\infty =(\hat{R}_1,\hat{R}_2, \hat{R}_3,\ldots ).\) 
For every \(n\ge 1\)  we define \(\varphi _{\infty }:S_{n+1}\rightarrow S_n\)  by \(\varphi _{\infty }=(\varphi ,\varphi ,\varphi ,\ldots )\)  and observe that by definition \(\varphi _{\infty }(\mathcal {P}_{n+1})=\mathcal {P}_n\) . Let for each \(n\ge 1\) , \(0_n\in \mathcal {P}_n\)  denotes inverse sequence formed by ramification points of \(n\) -ods \(T_n\) .
Clearly, each point \(x\in S_n\setminus \lbrace 0_n\rbrace \)  is \(2^n\) -periodic for \(\hat{R}_n\) .
We also let \(0_\infty =(0_n:n\ge 1 )\) .
Claim 4.3 
For every \(\varepsilon >0\)  there exists an \(\varepsilon \) -cover \(\mathcal {W}\)  of \(\mathcal {P}_n\)  by topological disks
such that if \(Z\subset \mathcal {P}_n\)  is a continuum and if \(d_H(Z,0_n)>\varepsilon \)  then there exists a chain subcover \(\mathcal {W}^{\prime }\subset \mathcal {W}\)  of \(Z\) .

It is enough to show that the lemma holds for \(\mathcal {P}_1\)  and then use the fact that \(\varphi \)  is a branched cover.
Fix \(\varepsilon >0\) . Let us cover the arc \(T_1=\lbrace (x,y):x\in [-1,1],y=0\rbrace \)  with a chain of disks \(\lbrace D_0,\ldots D_{j}\rbrace \)  in \(\mathbb {D}^2\)  so that \(T_1\in \textrm {int} (D_0\cup \ldots \cup D_{j})\) . For example we can set
\(D_0=\Bigg [-1-\frac{1}{j-1},-1+\frac{1}{j-1}\Bigg ]\times [2^{-j},-2^{-j}],\) 
\(D_1=\Bigg [-1+\frac{1}{j-1},-1+\frac{3}{j-1}\Bigg ]\times [2^{-j},-2^{-j}],\) 
\(\vdots \) 
\(\hspace{-9.95863pt}D_{j-1}=\Bigg [-1+\frac{2j-5}{j-1},-1+\frac{2j-3}{j-1}\Bigg ]\times [2^{-j},-2^{-j}],\) 
\(D_{j}=\Bigg [-1+\frac{2j-3}{j-1},-1+\frac{2j-1}{j-1}\Bigg ]\times [2^{-j},-2^{-j}].\) 
Now fix \(k\in \lbrace 0,...,j\rbrace \)  and \(i\in \mathbb {N}\)  and consider \(D_k^{\leftarrow }=\pi _{i,1}^{-1}(D_k)\) . Since the bonding maps \(\bar{f}_{1,1},\bar{f}_{2,1},...,\bar{f}_{i,1}\)  are near-homeomorphisms by [5]} they are also monotone. Hence, since \(D_k\)  is acyclic, the pre-image \(\bar{f}_{i,1}^{-1}\circ ...\circ \bar{f}_{1,1}^{-1}(D_k)\)  is acyclic as well. Indeed, since \(\mathbb {D}^2\setminus D_k\)  is connected so is \(\bar{f}_{i,1}^{-1}\circ ...\circ \bar{f}_{1,1}^{-1}(\mathbb {D}^2\setminus D_k)\) . Consequently \(D_k^{\leftarrow }\)  is also an acyclic continuum, since otherwise there would need to exist an \(i\)  such that \(\pi _{i,1}(D_k^{\leftarrow })\)  is not acyclic, leading to a contradiction.
Since \(\bar{f}_{i,1}\)  is a near-homeomorphism, choosing \(i\)  and \(j\)  large enough we can ensure that \(D_k^{\leftarrow }\)  has diameter less than \(\varepsilon /2\)  for each \(k\in \lbrace 0,\ldots ,j\rbrace \) . By Zoretti's Theorem [6]}, for any \(\delta <\varepsilon /2\)  there exists a simple closed curve \(C_k\)  around \(D_k^{\leftarrow }\) , each point of which is not farther away than \(\delta \)  from \(D_k^{\leftarrow }\) . Let \(\hat{C}_k\)  be the union of \(C_k\)  with its complementary domain containing \(D_k^{\leftarrow }\) . If \(\delta \)  is chosen small enough then \(\hat{C}_k\cap \hat{C}_{k^{\prime }}\ne \emptyset \)  if and only if \(D_k^{\leftarrow }\cap D_{k^{\prime }}^{\leftarrow }\ne \emptyset \)  for any \(k\ne k^{\prime }\) .
This completes the proof.
Claim 4.4 
\(\hat{R}_\infty |_{\mathcal {P}_\infty }\)  has a unique fixed point \(0_\infty \) , and for every \(z\in \mathcal {P}_\infty \setminus \lbrace 0_\infty \rbrace \)  we have that \(\hat{R}_\infty |_{\omega (\hat{R}_\infty ,z)}\) 
is a 2-adic odometer.

By definition \(0_\infty \)  is a fixed point of \(\hat{R}_\infty \) . If \(z\in \mathcal {P}_\infty \)  and \(z\ne 0_\infty \)  then \(z_n\)  is in a \(2^n\) -periodic orbit \(O_n\)  of \(\hat{R}_n\) ,
so \(z\)  is clearly a point in 2-adic odometer.
Claim 4.5 
\(\mathcal {P}_\infty =\varprojlim (\mathcal {P}_n,\varphi _\infty |_{\mathcal {P}_n}\rbrace \)  is a pseudo-arc.

Since each \(\mathcal {P}_n\)  is a pseudo-arc, the inverse limit of them \(\mathcal {P}_\infty \)  is also a pseudo-arc, as it must be both arc-like and hereditarily indecomposable, see [7]}.
The proof of Theorem REF  is now complete.
